The principles of adaptive noise compensation have been described by in Widrow B., Glover J. R., McCool J. M., Kaunitz J., Williams C. S., Heam R. H., Zeidler J. R., Dong jr. E., Goodlin R. C. Adaptive noise cancellation: principles and applications. Proc. IEEE, 63: 1692-1716, 1975. And incorporated herein by reference. This principal with multiple modifications and improvements have been widely used in a variety of applications.
The adaptive process consists of dynamically manipulating the vector of weighting factors W so that the expected value of the output signal c is minimized. This minimization process is often done using the Widrow-Hoff Least Mean Square (LMS) algorithm as described in the Windrow et al. publication identified above and in Chen J., Vanderwalle J., Sansen W., Vantrappen G., Janssens J. Adaptive method for cancellation of respiratory artifact in electrogastric measurements. Med. & Biol. Eng. & Comput, 27: 57-63, 1989 incorporated herein by reference.
Generally the primary signal is viewed as a summation of two signals, an information signal s1(t) and a noise signal n1(t). The secondary or reference signal consists of a noise signal n2(t) that is related to the noise component n1(t) in the primary signal. It is important to recognize that the two noise signals can differ in both phase and amplitude, but are considered strongly correlated. A compensation signal x(t) which under ideal conditions should be equal to the noise n1(t) is obtained through a combination of the reference signal, the output signal c(t) and the weighting factors w(t). Thus, by subtracting the compensation signal x(t) from the primary signal only the information signal s1(t) remains.
It is known that the adjustment of n2(t) for an optimal x(t) can be achieved by minimizing the energy of the output signal c(t) which is the difference between the primary (s1+n1) signal and the compensation signal x(t), thus: EQU c=s1+n1-x [1]
After squaring equation [1]: EQU c.sup.2 =s1.sup.2 +2s1(n1-x)+(n1-x).sup.2 [ 2]
one can subsequently calculate the expected value of the result (2): EQU E[c.sup.2 ]=E[s1.sup.2 ]+E[(n1-x).sup.2 ]+2E[s1(n1 -x)] [3]
Keeping in mind that E[s1.n1] and E[s1.x] correspond to the crosscorrelation between non-correlated signals: EQU 2E[s1(n1-x)]=0 [4]
Thus, equation [3] now becomes: EQU E[c.sup.2 ]=E[s1.sup.2 ]+E[(n1-x).sup.2 ] [5]
The goal of the adaptive compensation process is to minimize the second term of equation [5]: EQU min E[c.sup.2 ]=E[s1.sup.2 ]+min E[(n1-x).sup.2 ] [6]
Equation [6] represents the essence of Widrow-Hoff Least Mean Square (LMS) algorithm described in the Widrow et al. publication described above.
The minimal number of the weighting factors can be expressed with (10): EQU M=f.sub.max /f.sub.min [ 7]
where f.sub.max and f.sub.min are the frequencies of the maximal and minimal frequency components contained in the primary signal. However, it has been shown previously that this number should also be greater than 2f.sub.max see Sadasivan P. K. and Dutt D. N. A non-linear estimation model for adaptive minimization of EOG artifact from EEG signals. Int. J. on Bio-Medical Computing, 36:199-207, 1994. In the digital equivalent of adaptive filtering setup, all signals are usually presented with capital letters. The number of stored samples of the reference signal N2 and the number of weighting factors M is the same. The compensation signal X(j) is calculated as follows: ##EQU1## where j is the current sample, M is the number of weighting factors, N2.sub.k is the noise value after n samples, and W is the value of a given weighting factor.
For each new sample the weighting factors are recalculated based on their previous values, the reference signal N2 and its previous values, the output C and the feedback parameter .mu.. Using Widrow-Hoff's LMS algorithm the weighting factors can be determined with: EQU W.sub.ij+1 =W.sub.ij +2.mu.N2.sub.j-i C.sub.j [ 9]
where i ranges from j to j-M, j is the current sample, and W.sub.ij is the value of the i-th weighting factor in the j-th sample. By varying the feedback parameter .mu. the convergence speed and the accuracy of the adaptive filter can be manipulated. By setting .mu. higher the convergence speed is increased but accuracy lost (1, 2). Setting .mu. lower increases accuracy but slows convergence speed. The algorithm will remain stable if .mu. is maintained within the range: ##EQU2##
The adaptive system as described above is very effective for a variety of different applications, however it is not very effective when applied in non-ideal noise environments, which significantly limits the application to which the adaptive system may be applied.
Adaptive compensation based on the described Widrow-Hoff LMS algorithm provides reliable results only if two important conditions are met:
(1) the reference channel does not contain any information signal, and PA1 (2) the noise in the reference channel is strongly correlated with the noise in the primary channel. PA1 N3=a manipulated input derived from the error component N2 in reference signal R, and PA1 S3 is a manipulated input derived from the information component S2 of the reference signal R
In many real-life applications these two conditions are mutually contradictive--in order to provide the strongest possible correlation between the noise in the reference and in the primary channels the latter should be obtained from one and the same location. This, however, implies that the reference channel would contain also 100% of the information signal. At the other end of the scale, if the reference channel is obtained from a very remote location with respect to the primary channel so that the content of primary signal in the former is negligible (0%), the risk of reduction of the correlation relationship between the noises n1 and n2 increases significantly.
Kentie M. A., Van Der Schee E. J., Grashuis J. L., Smout A. J. P. M., (1981) Adaptive filtering of canine electrogastrographic signals. Part 1: system design. Med.& Biol. Eng. & Comput., 19, 759-764.(7) suggested a simple solution to this problem by modifying the adaptive compensator and deriving the reference from the primary channel. They were able to show better performance of the modified adaptive compensator as compared to an alternative bandpass filter with reversed frequency band and similar slope as the rejective filter used in the design. The improvement in the performance could be related to the reduction of the signal-to-noise ratio in the reference channel which possibly reduced the percentage of information signal in it. This method for adaptive filtering of information signals with broader frequency spectra makes it difficult, if not impossible to find an appropriate rejective filter to eliminate a significant number of frequency components related to the information signal alone which limits the applications of this.
Another suggested solution is to replace the noise in the reference channel with an artificially synthesized signal obtained with non-linear estimation using computer modeling as described in the hereinabove identified Sadasivan et al reference. This is difficult to do with acceptable accuracy particularly in environments with dynamic noise artifacts.
Under non-ideal applications where the information signal is present in the reference signal the adaptive filtering process will attempt to eliminate it in the primary signal, potentially resulting in distortions and/or decay of the output signal. The adaptive compensation technique applied in these circumstances would attempt to minimize the primary signal's data component.
None of the suggested solutions have proven to be effective for most non-ideal applications of adaptive compensation.